CASCADE PROTOCOL FOR iSWAP GATE IN A TWO-QUBIT SYSTEM

ABSTRACT

Methods, systems and apparatus for implementing iSWAP quantum logic gates between a first qubit and a second qubit. In one aspect, a method includes implementing a cascade schedule that defines a trajectory of a detuning between a frequency of the first qubit and a frequency of the second qubit. Implementing the cascade schedule includes: during a first stage, adiabatically driving detuning between the frequency of the first qubit and the frequency of the second qubit through a first avoided crossing in a leakage channel; during a second stage, driving detuning between the frequency of the first qubit and the frequency of the second qubit through a second avoided crossing in a swap channel; during a third stage, evolving the first qubit and second qubit; during a fourth stage, implementing the second stage in reverse order; and during a fifth stage, implementing the first stage in reverse order.

BACKGROUND

This specification relates to signal processing, quantum informationprocessing and condensed matter physics.

SUMMARY

This specification describes technologies for implementing high fidelityiSWAP logic gates in quantum computers.

In general, one innovative aspect of the subject matter described inthis specification can be implemented in a method for implementing aniSWAP quantum logic gate between a first qubit and a second qubit, themethod comprising: implementing a cascade schedule that defines atrajectory of a detuning between a frequency of the first qubit and afrequency of the second qubit, comprising: during a first stage,adiabatically driving detuning between the frequency of the first qubitand the frequency of the second qubit through a first avoided crossingin a leakage channel; during a second stage, driving detuning betweenthe frequency of the first qubit and the frequency of the second qubitthrough a second avoided crossing in a swap channel; during a thirdstage, evolving the first qubit and second qubit; during a fourth stage,implementing the second stage in reverse order; and during a fifthstage, implementing the first stage in reverse order.

Other implementations of this aspect include corresponding classical orquantum computer systems, apparatus, and computer programs recorded onone or more computer storage devices, each configured to perform theactions of the methods. A system of one or more computers can beconfigured to perform particular operations or actions by virtue ofhaving software, firmware, hardware, or a combination thereof installedon the system that in operation causes or cause the system to performthe actions. One or more computer programs can be configured to performparticular operations or actions by virtue of including instructionsthat, when executed by data processing apparatus, cause the apparatus toperform the actions.

The foregoing and other implementations can each optionally include oneor more of the following features, alone or in combination. In someimplementations the cascade schedule satisfies a local adiabaticevolution condition. The local adiabatic evolution condition may begiven by

Ψ_(g)(t)|∂_(t)Ψ_(e)(t)

=const·ω_(g)(t) or ∂_(t)θ(t)=const, where Ψ_(g)(t) and Ψ_(e)(t)represent instantaneous adiabatic eigenstates of an effectiveHamiltonian describing the leakage channel, θ(t) represents the controlangle, and ω_(g)(t)=√{square root over ((ϵ(t)−η₁)²+8g²)} represents atime dependent gap for the Hamiltonian describing the leakage channel,with ϵ(t) representing the detuning between the frequency of the firstqubit and the frequency of the second qubit, η₁ representing ananharmonicity parameter of the first qubit, and g representinginterqubit interaction strength.

In some implementations the probability that the cascade schedule incursa leakage error is proportional to

$e^{{- 2}\sqrt{2}{g \cdot t_{p}}}$

with g representing the interqubit interaction strength and t_(p)representing a total duration of the iSWAP gate.

In some implementations the cascade schedule synchronizes minimal errorsin the swap channel and leakage channel.

In some implementations the first qubit and second qubit comprisecapacitively coupled Xmon qubits.

The leakage channel may comprise a manifold spanned by the computationalstate 11 and two non-computational states 02 and 20, and wherein drivingdetuning between the frequency of the first qubit and the frequency ofthe second qubit through a first avoided crossing in a leakage channelcomprises driving detuning between the frequency of the first qubit andthe frequency of the second qubit through state 11-20 resonance.

The swap channel may comprise a manifold spanned by the computationalstates 10 and 01, and wherein driving detuning between the frequency ofthe first qubit and the frequency of the second qubit through a secondavoided crossing in a swap channel comprises driving detuning betweenthe frequency of the first qubit and the frequency of the second qubitthrough state 10-01 resonance.

In some implementations implementing the second stage in reverse ordercomprises driving detuning between the frequency of the first qubit andthe frequency of the second qubit to achieve a complete population swapbetween the qubit states 10 and 01.

In some implementations the control angle comprises an angle between aneffective magnetic field and the z-axis on a Bloch sphere of a systemcomprising the first qubit and the second qubit.

In some implementations the method may further comprise generating thecascade schedule, comprising: defining a trapezoidal ramp function inlaboratory time; generating a polynomial expansion of the waveform forthe control angle in terms of the generated trapezoidal ramp function;generating an initial schedule for driving the detuning between thefrequency of the first qubit and the frequency of the second qubit usingthe generated polynomial expansion of the waveform for the controlangle; obtaining one or more solutions in the swap channel; determiningthe probability of an error in the leakage channel; and adjusting theinitial schedule for driving the detuning by synchronizing the one ormore solutions in the swap channel and the probability of an error inthe leakage channel, comprising: selecting a solution in the swapchannel that, when combined with the probability of an error in theleakage channel, achieves a predetermined swap gate fidelity; andapplying values of parameters in the selected solution to the initialschedule for driving the detuning to generate the cascade schedule.

In some implementations the method may further comprise adjustingparameter values of the cascade schedule using gate fidelity as afitness function.

In some implementations the method may further comprise performinghardware testing and randomized benchmarking techniques to adjust thegenerated cascade schedule.

In some implementations the trapezoidal ramp function is a function ofmaximum control angle, pulse duration, length of upward ramp, and lengthof downward ramp.

In some implementations the trapezoidal ramp function takes values thatare less than or equal to one.

In some implementations coefficients of the polynomial expansion sum toone.

In some implementations generating the initial schedule optionallyfurther comprises applying a Gaussian filter.

In some implementations generating the initial schedule comprisesparameterizing the detuning as ϵ(t)=μ+2g·λ cot[Θ(t,{c})] with μrepresenting shift, λ representing scaling, g representing interqubitinteraction strength, Θ representing the generated polynomial expansionand c representing a set of two or more variational parameters.

In general, another innovative aspect of the subject matter described inthis specification can be implemented by an apparatus comprising a firstqubit, a second qubit coupled to the first qubit, and controlelectronics comprising one or more control devices that tune thefrequency of the first qubit and second qubit through application ofrespective control signals, wherein the control electronics areconfigured to implement the cascade schedule.

The subject matter described in this specification can be implemented inparticular ways so as to realize one or more of the followingadvantages.

Execution of iSWAP gates using known protocols is slow and susceptibleto leakage. In particular, applications of known techniques, such assimple trapezoidal pulses, introduce leakage errors. The presentlydescribed techniques for executing iSWAP gates provide fast and robustcascade schedules in the well-defined adiabatic regime that allow for acomplete SWAP operation accompanied by a suppression of the Leakageerror. For example, some implementations result in a suppression of theleakage error to below 10⁻⁴ in a very broad region of the gate times 13ns, which in turn leads to the estimated fidelity of the iSWAP gateexceeding 99.9999% and the fidelity loss due to a time error˜5·10⁻⁵ nsin the vicinity of the optimal gate time t_(p)=46.8 ns.

In addition, the presently described techniques for executing iSWAPgates utilize only low-frequency control of detuning between qubitfrequencies and are applicable to near term quantum computingarchitectures. Furthermore, since iSWAP gates are computationally hardto simulate using classical computers, the techniques described in thisspecification can facilitate state of the art experiments on quantumsupremacy and an immediate impact on the field of quantum computing.

Implementation of the presently described techniques can reduce circuitdesign complexity and provide a path to scalable quantum computingarchitectures with high-fidelity multi-qubit gates. This, in turn, is animportant step forward in achieving the long-term goal of developing anerror-corrected quantum computer.

Details of one or more implementations of the subject matter of thisspecification are set forth in the accompanying drawings and thedescription below. Other features, aspects, and advantages of thesubject matter will become apparent from the description, the drawings,and the claims.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 depicts an example system for implementing iSWAP quantum logicgates.

FIG. 2 shows a plot comparing an example adiabatic protocol forperforming an iSWAP gate and an example non-adiabatic protocol forperforming an iSWAP gate.

FIG. 3 shows two plots of the energy levels of two coupled Xmon qubitssubject to a non-adiabatic protocol.

FIG. 4 shows two plots of the energy levels of two coupled Xmon qubitssubject to an adiabatic protocol.

FIG. 5 shows a first plot comparing a Rabi protocol and a Rosen-Zenerprotocol and a second plot comparing error transition probabilities ofthe Rabi protocol and the Rosen-Zener protocol.

FIG. 6 is a flow diagram of an example process for implementing an iSWAPquantum logic gate between a first qubit and a second qubit according toa cascade schedule.

FIG. 7 shows two example cascade schedules.

FIG. 8 shows a plot of the probabilities of interqubit population swapfor two example cascade schedules.

FIGS. 9 and 10 show example synchronizations of leakage and SWAP errors.

DETAILED DESCRIPTION

A complete SWAP operation is a trace preserving, i.e., zero-leakage,transformation of a two-qubit system that enables complete populationtransfer between the states |1

⊗|0

and |0

⊗|1

(herein written as states 10 and 01) of the SWAP channel. Mostgenerally, such an operation can be described by the unitary matrix

${U = \begin{pmatrix}1 & 0 & 0 & 0 \\0 & 0 & {- {ie}^{i\;\theta_{1}}} & 0 \\0 & {- {ie}^{i\;\theta_{2}}} & 0 & 0 \\0 & 0 & 0 & {e^{i{({\theta_{1} + \theta_{2}})}}e^{i\;\phi}}\end{pmatrix}},$

in the (|00

|01

|10

|11

) basis, where represents a phase shift and angles θ₁ and θ₂ representadditional phases that can be corrected using single qubit z-rotations.Since these additional phases can be disregarded the SWAP and iSWAPgates can be described by the unitary matrices given in equation (1)below

$\begin{matrix}{{U_{swap} = \begin{pmatrix}1 & 0 & 0 & 0 \\0 & 0 & {- 1} & 0 \\0 & {- 1} & 0 & 0 \\0 & 0 & 0 & e^{i\;\phi}\end{pmatrix}},{U_{iswap} = \begin{pmatrix}1 & 0 & 0 & 0 \\0 & 0 & {- i} & 0 \\0 & {- i} & 0 & 0 \\0 & 0 & 0 & e^{i\;\phi}\end{pmatrix}}} & (1)\end{matrix}$

A complete population swap in the |01)-|10) channel is referred to as a“perfect” swap operation, e.g., a swap operation as realized in theory.A near perfect or almost perfect swap operation is a swap operation withswap and leakage errors below a predetermined threshold, e.g., errors onthe order of 10⁻⁵. Other thresholds can also be used. For example, anerror threshold can be selected based on a target fidelity specified byoperators of the quantum computer device.

A challenging aspect of implementing SWAP or iSWAP gates is that theimplementation should be consistent with generic quantum computingarchitectures, e.g., capacitively coupled qubits. In such architectures,qubit frequencies are parked in a so-called zigzag order, withfrequencies of the nearest-neighbor qubits separated by ˜1 GHz. Aftergate operations are applied to the qubits, the qubits are returned totheir original parking positions. This makes some protocols, e.g.,direct passage driving schedules of the Landau-Zener type, difficult orimpossible to apply.

In addition, execution of a SWAP or iSWAP gate between a first qubit andsecond qubit by driving detuning ε(t) between energy levels of 10 and 01states to zero (10-01 resonance), inevitably requires passing through a11-20 leakage resonance first. As a result, a simple, trapezoidal-shaperamp function is not sufficient to reliably avoid leakage intonon-computational sector of the Hilbert space. Existing hardwareimplementations of the SWAP or iSWAP gates also suffer from slowexecution.

The techniques described in this specification address these challenges.In particular, this specification describes cascade protocols that canbe applied to qubits to execute two-qubit gates such as SWAP or iSWAPgates with improved fidelities. The described cascade protocols utilizelow-frequency control of detuning between qubit frequencies. Morespecifically, the described techniques employ a multi-parameter set ofbias-controlled time protocols that are configured to suppress botherrors in the SWAP and Leakage quantum channels at the end of gateexecution. The shape and duration of the defined pulse is obtained usingsynchronization of motion in both channels, which results in an almostperfect SWAP operation accompanied by a reduction of probability toexcite individual qubits into higher-frequency anharmonic states withboth the swap and the leakage errors on the order of 10⁻⁵.

For convenience, the techniques described in this specification arepresented with reference to implementing iSWAP gates. However, thetechniques can be equally applied to implementing SWAP gates, as well asother gates that are based on SWAP operations, e.g. a √{square root over(SWAP)} gate.

Example Architecture

FIG. 1 depicts an example system 100 for implementing an iSWAP quantumlogic gate on a two-qubit subsystem. The example system 100 is anexample of a system implemented as part of a quantum computing device inwhich the systems, components and techniques described in thisspecification can be implemented.

The system 100 includes a two-qubit subsystem 102 in communication withcontrol electronics 104. The two-qubit subsystem 102 includes a firstqubit 106 and a second qubit 108. In some cases, as shown in FIG. 1, thefirst qubit 106 and second qubit 108 may be capacitively coupled Xmonqubits. For example, the first qubit 106 and second qubit 108 may bepart of a linear chain of Xmon qubits included in a quantum computingdevice. However, in other cases the qubits may include flux qubits,phase qubits, or qubits with frequency interactions.

The first qubit 106 and second qubit 108 can be operated by adjustingthe qubit frequencies, e.g., applying pulses generated by the controlelectronics 104 to the qubits. In cases where the first qubit 106 andthe second qubit 108 are Xmon qubits, the qubit frequencies may beparked at a predetermined distance from one another, and in a zig-zagposition with respect to other qubits that may be included in thequantum computing device.

A Hamiltonian describing the two qubits may be given by equation (2)below.

$\begin{matrix}{H = {{\sum\limits_{i = 1}^{2}\;\left\lbrack {{\omega_{i}(t)} - {\frac{1}{2}\eta_{i}{n_{i}\left( {n_{i} - 1} \right)}}} \right\rbrack} + {g\left( {{a_{1}^{\dagger}a_{2}} + {a_{2}^{\dagger}a_{1}}} \right)}}} & (2)\end{matrix}$

In equation (2), ω_(i)(t) represents time-dependent natural frequenciesof the individual qubits, η_(i) represents anharmonic detunings, grepresents interqubit interaction strength, and η_(i)=a_(i) ^(†)a_(i)represents the number operator and a_(i) ^(†), a_(i) represent creationand annihilation operators. Without loss of generality, η₁=η₂=η, withtypical values of η=2π×200 MHz and g=2π×20 MHz. Furthermore, it can beassumed that ω₂=const and ω₁(t)=ω₂+ε(t), where ε(t) represents thecontrolled detuning with the initial and final valuesε(−t_(p)/2)=ε(t_(p)/2)=2π×1 GHz, and t_(p) represents the duration of apulse applied to the qubit or qubits to implement a quantum logic gate(the gate time).

The Hamiltonian described in equation (2) conserves the total number ofexcitations M. Therefore, the 9×9 Hilbert space splits into 5 subspacescorresponding to M=0, 1 . . . , 4. Three of these subspaces with M=0, 1,2 are relevant for the qubit operations driven by ε(t). These are theground state 00, the SWAP manifold spanned by the computational states10 and 01 and the leakage manifold spanned by the computational state 11and two non-computational states 02 and 02. Sub-Hamiltonians H_(s)(t)and H_(l)(t) describing the SWAP channel and the leakage channel cantherefore be expressed as

$\begin{matrix}{{{H_{s}(t)} = {\omega_{2} + \begin{pmatrix}{\epsilon(t)} & g \\g & 0\end{pmatrix}}}{{H_{l}(t)} = {{2\omega_{2}} + {\begin{pmatrix}{\epsilon(t)} & {\sqrt{2}g} & {\sqrt{2}g} \\{\sqrt{2}g} & {{2{\epsilon(t)}} - \eta_{1}} & 0 \\{\sqrt{2}g} & 0 & {- \eta_{2}}\end{pmatrix}.}}}} & (3)\end{matrix}$

All parameters except ε(t) of the Hamiltonian in equation (2) arestatic—they are not changed during execution of a quantum logic gate.Therefore, a schedule for implementing a iSWAP gate is achieved throughparametrization of the detuning ε(t). The detuning may take the form

ε(t)=μ+2g·λ cot[ϑ(t,{c})]  (4)

where the control angle ϑ depends on a set of variational parameters {c}containing M≥2 elements and the two additional parameters μ (shift) andλ (scaling) may be used in an extended adjustment/optimizationprocedure.

The parameters μ and λ define two limiting cases known in the art asnon-adiabatic (μ=0, λ=1) and adiabatic (μ=η₁, λ=√{square root over (2)})protocols, respectively. The differences between these two protocols areillustrated in FIGS. 2-4.

The control electronics 104 include control devices, e.g., arbitrarywaveform generators, that can operate the first qubit 106 and secondqubit 108. For example, the control electronics 104 may include controldevices that tune the frequency of the first qubit 106 and second qubit108 by applying control signals, e.g., voltage pulses, to the qubitsthrough respective control lines.

In addition, the control devices may include measurement devices, e.g.,readout resonators, that can perform measurements of the first qubit 106and the second qubit 108 through respective qubit control lines. Thecontrol electronics 104 may be configured to store, display, and/orfurther process the results of measurements of the first qubit 106 andthe second qubit 108.

In some implementations, the control electronics 104 may include a dataprocessing apparatus and associated memory. The memory may include acomputer program having instructions that, when executed by the dataprocessing apparatus, cause the data processing apparatus to perform oneor more functions described herein, such as applying a control signal toa qubit.

FIG. 2 shows a plot 200 comparing an example adiabatic protocol 202 forperforming an iSWAP gate and an example non-adiabatic protocol 204 forperforming an iSWAP gate. The plot includes a horizontal axis 206representing dimensionless time 2t/t_(p) (where t_(p) represents gatetime) and a vertical axis 208 representing the detuning ϵ(t)/2π measuredin GHz. A first horizontal line 210 defines points of the levelcrossings in the SWAP channel. A second horizontal line 212 definespoints of the level crossings in the leakage channel.

The distinction between the adiabatic protocol 202 and non-adiabaticprotocol 204 is clear when the slopes of ε(t) in the leakage channel 212near 11-20 resonance (avoided level crossing), which occurs at ε=η₁, arecompared. The non-adiabatic protocol 204 passes through the crossing 212with very large velocity while the adiabatic protocol 202 has aninflection point corresponding to the minimal relative velocity of theenergy levels. For the SWAP channel 210 the 10-01 resonance occurs atε=0 and the behavior is the opposite, i.e. the non-adiabatic protocol204 has an inflection point while the adiabatic protocol 202 dropsalmost vertically.

As shown in FIG. 2, the adiabatic protocol 202 takes the shape of a“cascade waterfall” that drops quickly and slows down near thehorizontal line 212, i.e. it forms a “ledge” and then drops quickly andforms a ledge again. This behavior reflects the idea of a localadiabatic evolution for a system with several level crossings. In otherwords, the schedule behaves as a cascade, it slows down near eachavoided crossing and accelerates again after passing it. Thenon-adiabatic protocol 204 goes straight down as a “plunge waterfall”.

The energy eigenvalues of the Hamiltonian given by Equation (1) in theleakage channel and SWAP channel are shown in FIGS. 3 and 4 fornon-adiabatic and adiabatic protocols, respectively.

FIG. 3 shows two plots 300, 350 of the energy levels of two coupled Xmonqubits subject to a non-adiabatic protocol. The first plot 300 showsenergy levels in the leakage manifold. The second plot 350 shows energylevels in the SWAP manifold. Both plots include a horizontal axisrepresenting dimensionless time 2t/t_(p) (where t_(p) represents gatetime) and a vertical axis representing the energy level measured in GHz.

FIG. 4 shows two plots 400, 450 of the energy levels of two coupled Xmonqubits subject to an adiabatic protocol. The first plot 400 shows energylevels in the leakage manifold. The second plot 450 shows energy levelsin the SWAP manifold. Both plots include a horizontal axis representingdimensionless time 2t/t_(p) (where t_(p) represents gate time) and avertical axis representing the energy level measured in GHz.

Proposed iSWAP Gate Schedule

The proposed protocol for implementing an iSWAP gate between a firstqubit and a second qubit includes a cascade schedule that defines atrajectory of a detuning between a frequency of the first qubit and afrequency of the second qubit. The cascade schedule includes multiplestages: a two stage ramp-down passage, a plateau stage, and a two-stageramp-up passage in reverse order that preserves an overall time reversalsymmetry of the protocol.

The two-stage ramp-down passage includes a first stage where thedetuning between the frequency of the first qubit and the frequency ofthe second qubit is adiabatically driven through the avoided crossing inthe leakage channel (ϵ(t)=η) to avoid leakage error, and a second stagewhere the detuning between the frequency of the first qubit and thefrequency of the second qubit is driven through the avoided crossing inthe SWAP channel (ϵ(t)=0) to achieve a complete population swap.

During the plateau stage (third stage) the first qubit and the secondqubit are allowed to freely evolve for typical times ˜10 ns.

The two-stage ramp-up passage includes a fourth stage where the secondstage described above is implemented in reverse order to achieve acomplete population swap in the swap channel, and a fifth stage wherethe first stage is implemented in reverse order to avoid leakage error.The fourth and fifth stages preserve time reversal symmetry. Examplecascade schedules for implementing iSWAP gates between a first andsecond qubit are described in further detail below with reference toFIG. 7.

As described below in relation to FIG. 6, the cascade schedule isgenerated based on a trapezoidal waveform for the control angle θ(t)defining a trajectory of a frequency of the first qubit during executionof the protocol. The motion of the control vector during the proposedprotocol decelerates in the middle of the ramp-down process (end of thefirst stage) and then accelerates again (beginning of the fifth stage).The deceleration near the avoided crossing of the leakage channel isgoverned by an adiabatic Rosen-Zener schedule, as described below, whichresults in the exponential decay of the leakage error in a broad rangeof the gate times.

The proposed cascade schedule for the iSWAP gate described in thisspecification satisfies a condition of local adiabatic evolution. Thiscondition means that detuning between the frequency of the first qubitand frequency of the second qubit may change quickly far away fromavoided crossings, i.e. for the frequencies exceeding 100 MHz distancefrom the point of the avoided crossing (above or below), and mustslowdown in proximity to the minimum gap between energy levels, i.e.within the frequency interval ˜200 MHz centered the avoided crossingfrequency.

Implementing iSWAP Gates

The proposed schedule is designed to utilize adiabatic motion in theleakage channel and reduce leakage error in a broad region of gateexecution time. Motivation for the design of the proposed schedule canbe illustrated by considering a simplified example. Design of theproposed schedule begins by considering an effective two-level leakageHamiltonian of equation (3) describing only |11

and |20

states:

$\begin{matrix}{{{\overset{\_}{H}}_{l}(t)} = \begin{pmatrix}{\epsilon(t)} & {\sqrt{2}g} \\{\sqrt{2}g} & {{2{\epsilon(t)}} - \eta_{1}}\end{pmatrix}} & (5)\end{matrix}$

This Hamiltonian can also be represented as

$\begin{matrix}{{{{\overset{\_}{H}}_{l}(t)} = {{- \frac{\omega_{g}(t)}{2}}\begin{pmatrix}{\cos\mspace{14mu}{\theta(t)}} & {\sin\mspace{14mu}{\theta(t)}} \\{\sin\mspace{14mu}{\theta(t)}} & {{- \cos}\mspace{14mu}{\theta(t)}}\end{pmatrix}}},} & (6)\end{matrix}$

where ω_(g)(t)=√{square root over ((ϵ(t)−η₁)²+8 g²)} represents thetime-dependent gap in the leakage channel, g represents interqubitinteraction strength, the parameter

$\begin{matrix}{{\theta(t)} = {{arccot}\left\lbrack \frac{{\epsilon(t)} - \eta_{1}}{2\sqrt{2}g} \right\rbrack}} & (7)\end{matrix}$

represents the control angle between the control vector, i.e. effectivemagnetic field b(t)=(2√{square root over (2)}g,0, ϵ(t)−η₁), and thez-axis on a Bloch sphere related to the two-dimensional Hilbert spacespanned by the diabatic states |11

and |20

. The control angle defines the motion of the first qubit whosefrequency is varied to drive detuning during execution of the iSWAPgate.

The proposed cascade schedule for the iSWAP gate described in thisspecification satisfies a condition of local adiabatic evolution. Thiscondition means that detuning between the frequency of the first qubitand frequency of the second qubit may change quickly far away fromavoided crossings, i.e. for the frequencies exceeding 100 MHz distancefrom the point of the avoided crossing (above or below), and mustslowdown in proximity to the minimum gap between energy levels, i.e.within the frequency interval ˜200 MHz centered the avoided crossingfrequency.

As such, the condition of the local adiabatic evolution can be broadlydefined. A particular form of the local adiabatic condition that can beutilized in the proposed schedule is given by

ψ_(g)(t)|∂_(t)ψ_(e)(t)

=const·ω_(g)(t)  (8)

where ψ_(g)(t) and ψ_(e)(t) represent instantaneous ground and excitedadiabatic eigenstates of the leakage Hamiltonian (6), respectively.

Analysis of Alternative Forward Single-Passage Schedule

Alternative methods for implementing an iSWAP gate satisfying the localadiabatic condition given by equation (8) includes implementation of aforward single-passage schedule where detuning sweeps a large energyfrom ϵ_(i)>0 to ϵ_(f)<0 and a system including the first qubit andsecond qubit passes the avoided crossing only once. Since the qubits donot return to their original parked states this schedule cannot bedirectly used in Xmon architecture, but may be used in otherarchitectures that are not constrained in this way.

The local adiabatic condition given by equation (8) and the definitionof the angle θ(t) given by equation (7) imply that ∂_(t)θ(t)=const, or

$\begin{matrix}{{\theta(t)} = {\frac{\pi}{2}\left( {1 + \frac{2t}{t_{p}}} \right)}} & (9)\end{matrix}$

Equation (9) shows that the control vector uniformly rotates within x,z-plane starting from the North pole at t=−t_(p)/2 and ending at theSouth pole t=t_(p)/2 where t_(p) represents the gate execution time.However, the magnitude of the control vector is not zero and has verysharp time dependence. As a result, the problem cannot be reduced to atime-independent Hamiltonian. Fortunately, for the time dependence givenby equation (9) the problem can be solved exactly using transformationof the time-dependent Schrödinger equation to the natural time scale:

τ(t)=ω₀ ⁻¹∫₀ ^(t)ω_(g)(s)ds,  (10)

where ω₀=2√{square root over (2)}g represents the minimal gap betweenthe (adiabatic) eigenstates of Hamiltonian (6).

Equation (10) defines a one-to-one map between the “ordinary” (or“laboratory”) time tin the original Hamiltonian (6) and an artificialtime variable τ describing motion in an accelerating frame of referencein which the magnitude of the effective magnetic field (control vector)is a time-independent constant. In other words, in the natural timeframe the Hamiltonian of a two-level system has a constant gap ω₀ andthe motion of the system is determined by the dependence of the controlangle on the natural time θ(τ)=θ[t(τ)]. Physically, ω₀τ(t) representsthe phase accumulated by the first qubit during the time interval (0,t).

The time-dependent Schrödinger equation for the leakage Hamiltonian inequation (6) is given by:

$\begin{matrix}{{i{\partial_{t}\begin{pmatrix}{\Psi_{0}(t)} \\{\Psi_{1}(t)}\end{pmatrix}}} = {{{\overset{\_}{H}}_{l}(t)}\begin{pmatrix}{\Psi_{0}(t)} \\{\Psi_{1}(t)}\end{pmatrix}}} & (11)\end{matrix}$

where Ψ₀(t) and Ψ₁(t) represent the projection of the time-dependentstate vector onto the original (time-independent) basis states |11

and |20

. In the nomenclature adapted throughout this specification the diabaticbasis states of a generic two-level system are labeled with a pair ofindices (0,1), e.g. |0

≡|11

and |1

≡|20

in the present case or |0

≡|10

and |1

≡01

for the SWAP channel, while the adiabatic basis states, i.e.time-dependent eigenstates of the Hamiltonian H _(l)(t) in the presentcase or H_(s)(t) for the SWAP channel are labeled with a pair of indices(g,e). The solution of equation (11) can be obtained usingtransformation to instantaneous adiabatic basis of the Hamiltonian H_(l)(t) defined by the time-dependent unitary matrix:

$\begin{matrix}{{U(t)} = {{{\sin\left\lbrack \frac{\theta(t)}{2} \right\rbrack}\sigma_{x}} + {{\cos\left\lbrack \frac{\theta(t)}{2} \right\rbrack}\sigma_{z}}}} & (12)\end{matrix}$

where σ_(i) are the Pauli matrices. Applying the unitary matrix U(t) toboth sides of equation (11) and performing the natural timetransformation (9) results in the transformed Schrödinger equation inthe adiabatic basis and in the natural time scale:

$\begin{matrix}{{i{\partial_{\tau}\begin{pmatrix}{\psi_{g}(\tau)} \\{\psi_{e}(\tau)}\end{pmatrix}}} = {\frac{1}{2}\begin{pmatrix}{- \omega_{0}} & {{- i}\;{\theta^{\prime}(\tau)}} \\{i\;{\theta^{\prime}(\tau)}} & \omega_{0}\end{pmatrix}\begin{pmatrix}{\psi_{g}(\tau)} \\{\psi_{e}(\tau)}\end{pmatrix}}} & (13)\end{matrix}$

Here ψ_(g,e)(τ)≡ψ_(g,e)[t(τ)] represents the components of the adiabaticwave function expressed through the natural time, and t(τ) is thesolution of equation (10) for t. Equation (13) describes the motion ofthe two-level system with a constant gap ω₀ and non-adiabatic couplingθ′(τ).

The substitution ψ_(g)(τ)=e^(iω) ⁰ ^(τ/2)c_(g)(τ) and ψ_(e)(τ)=e^(−iω) ⁰^(τ/2)c_(e)(τ) into equation (13) leads to the system of equations forc_(g,e)(τ) given by

$\begin{matrix}{{c_{g}^{\prime}(\tau)} = {{- \frac{1}{2}}e^{{- i}\;\omega_{0}\tau}{\theta^{\prime}(\tau)}{c_{e}(\tau)}}} & \left( {14a} \right) \\{{c_{e}^{\prime}(\tau)} = {\frac{1}{2}e^{i\;\omega_{0}\tau}{\theta^{\prime}(\tau)}{c_{g}(\tau)}}} & \left( {14b} \right)\end{matrix}$

This, in turn, can be transformed into the second order differentialequation for the coefficient c_(e)(τ):

$\begin{matrix}{{{c_{e}^{''}(\tau)} - {\left\lbrack {{i\;\omega_{0}} + \frac{\theta^{''}(\tau)}{\theta^{\prime}(\tau)}} \right\rbrack{c_{e}^{\prime}(\tau)}} + {\frac{1}{4}{\theta^{\prime}(\tau)}^{2}{c_{e}(\tau)}}} = 0} & (15)\end{matrix}$

It is assumed that at the initial moment τ_(in) the system is in theground state, i.e. ψ=_(e)(τ_(in))=ψ_(e)(−t_(p)/2)=0, giving the initialcondition c_(e)(τ_(in))=0. Since c_(g)(τ_(in))=1 is known, the secondboundary condition for the derivative c′_(e)(τ_(in)) is given byequation (14b). The leakage error is given by the probability that thesystem will populate the excited state at the end of the executionτ_(f):

P _(e)=|ψ_(e)(t _(p)/2)|²=ψ_(e)(τ_(f))|² =|c _(e)(τ_(f))|².  (16)

Here the natural times τ_(in) and τ_(f) correspond to the beginning andthe end of the gate and can be obtained from equation (10), whichprovides the map (−t_(p)/2,t_(p)/2)⇔(τ_(in),τ_(f)).

From equations (7), (9) and (10) the explicit relation between the timest and τ, and, ultimately, the functional dependence θ(T) can be found:

$\begin{matrix}{{\theta(\tau)} = {{\theta\left\lbrack {t(\tau)} \right\rbrack} = {{\frac{1}{2}\left\lbrack {\pi + {4{\arctan\left( {\tanh\left( \frac{\pi\tau}{2t_{p}} \right)} \right)}}} \right\rbrack}.}}} & (17)\end{matrix}$

Equations (15) and (17) can be recognized as the Rosen-Zener model,which was developed to describe a double Stern-Gerlach experiment. Byinitial design, the problem describes a process evolving during aninfinite time interval. The Rosen-Zener problem can be solved exactly interms of hypergeometric functions. Protocols related to this typesolution are referred to herein as “Rosen-Zener” protocols. The samesolution presented below can be obtained for the finite pulse timet_(p). This is because by virtue of equations (7) and (9), thetime-dependent gap is given by ω_(g)(t)=ω₀ sec(πt/t_(p)), and theintegral in equation (10) diverges at t=±t_(p)/2, i.e. finite gate timet_(p) maps onto the infinite time in the natural time scale: (−t_(p)/2t_(p)/2)⇔(−∞,∞).

The solution of equation (15) with θ(τ) given by equation (17) can beobtained by replacing τ with a new variable, z(τ)=(½)[1+tanh(πτ/t_(p))],which changes from 0 to 1 when τ changes from −∞ to ∞. After the changeof variables equation (15) takes the form that can be recognized as ahypergeometric equation

$\begin{matrix}{{{{{z\left( {1 - z} \right)}{c_{e}^{''}(z)}} - {\left( {z - \frac{1}{2} + {i\frac{\gamma}{\pi}}} \right){c_{e}^{\prime}(z)}} + {\frac{1}{4}{c_{e}(z)}}} = 0},} & (18)\end{matrix}$

which possess a general solution given by

$\begin{matrix}{{c_{e}(z)} = {{A_{1} \cdot {{{}_{}^{}{}_{}^{}}\left( {{- \frac{1}{2}},{\frac{1}{2};{\frac{1}{2} - {i\frac{\gamma}{\pi}}};z}} \right)}} + {{A_{2} \cdot z^{{1\text{/}2} + {i\;\gamma\text{/}\pi}}}{{{{}_{}^{}{}_{}^{}}\left( {{1 + {i\frac{\gamma}{\pi}}},{{i\frac{\gamma}{\pi}};{\frac{3}{2} + {i\frac{\gamma}{\pi}}};z}} \right)}.}}}} & (19)\end{matrix}$

Here ₂F₁(p,q;r;z) represents hypergeometric functions, γ=ω₀t_(p)/2represents the dimensionless gate time and the constants A_(1,2) must bedetermined from the initial conditions. The first initial condition atz=0, c_(e)(0)=0, implies that A₁=0, and the second boundary conditiongiven by equation (14b) allows determining A₂=(1+2iγ/π)⁻¹. As a result,the leakage error is given by a simple analytical formula:

P _(e)(γ)=|c _(e)(1)|²=sech²(γ)  (20)

Equation (20) provides an important physical insight behind the designof the proposed protocol. As can be seen from equation (20), theprobability that the cascade schedule incurs a leakage error isproportional to 4e^(−2γ)=4e^(−ω) ⁰ ^(t) ^(p) and decays exponentially atlarge times t_(p). This behavior is unique since, usually, the finitetime protocols result in a power-law decay of the non-adiabatic error.This can be illustrated by considering a different local adiabaticcondition as compared with equation (8). For example the condition

ψ_(g)(t)|∂_(t)ψ_(e)(t)

=const·ω_(g) ²(t) also leads to the exactly solvable problem. It can beshown that the solution can be obtained by means of the same naturaltime transformation described above, which results in thetime-independent Hamiltonian and the analytical formula for the leakageerror:

$P_{e} = {\frac{1}{\gamma^{2} + 1}{{\sin^{2}\left( {\frac{\pi}{2}\sqrt{\gamma^{2} + 1}} \right)}.}}$

The solution is formally similar to that of the known Rabi problemdescribing a spin moving under a uniformly rotating in-plane magneticfield of constant magnitude, and protocols related to solutions of thistype are therefore referred to herein a “Rabi protocol”. The power-lawdecay of the leakage error, P_(e)˜(ω₀t_(p))⁻², is slow as compared withthe fast exponential decay germane to the Rosen-Zener protocol.

FIG. 5 shows two plots 500 and 550. The first plot 500 compares a Rabiprotocol for implementing an iSWAP gate and a Rosen-Zener protocol. Thesecond plot 550 compares the error transition probabilities of the Rabiprotocol and the Rosen-Zener protocol.

Plot 500 includes a horizontal axis representing laboratory time tdivided by gate time t_(p), and a vertical axis representing detuningϵ(t)/2π measured in GHz. Lines 502 and 504 represent the Rabi protocol.Lines 506 and 508 represent the presently described Rosen-Zenerprotocol. Dashed lines 504 and 508 represent adiabatic protocols. Solidlines 502 and 506 represent non-adiabatic protocols.

Plot 550 includes a horizontal axis representing dimensionless totalduration of the iSWAP gate γ and a vertical axis representing theprobability that the cascade schedule incurs a SWAP or leakage errorP_(e)(γ). Line 552 represents the Rabi protocol. Line 554 represents thepresently described Rosen-Zener protocol.

As shown in FIG. 5, the calculated transition probabilities for the Rabiand Rosen-Zener protocols differ. While the Rabi protocol has welldefined zeros that would correspond either to a complete SWAP operationor to zero leakage, the overall behavior of the probability is highlynon-adiabatic because it decays slowly, i.e. proportional to1/(g·t_(p))² for large t. On the other hand, the Rosen-Zener scheduleprobability does not reach zero but approaches zero exponentially fast.In this sense the Rosen-Zener pulse shape is the “most adiabaticschedule” since it decays faster than any power of (ω₀t_(p))⁻¹. Thisfact provides physical motivation for the proposed protocol.

Method for Generating the Proposed “Rosen-Zener” Cascade Schedule

In what follows the time-independent diabatic basis 0 and 1 associatedwith the eigenstates of the Hamiltonian at initial time when the levelsare very far from each other are used, since it is more convenient fornumerical implementation. By the same token only return schedulescompatible with Xmon architecture are considered and the gatesimulations are performed within a time interval (0,t_(p)). The mainquantity of interest, P₀₁(t_(p)), is the probability of the transitionfrom the initial state 0 to the final state 1 at the end of the gate,t=t_(p). In the SWAP channel P₀₁(t_(p)) describes the probability of thepopulation swap between the computational states |10

and |01

. As such, P₀₁(t_(p)) is the probability of successful SWAP operationand P_(s)=1−P₀₁(t_(p)) represents the SWAP error. On the contrary, inthe leakage channel, P₀₁ is the probability of a transition from thecomputational state |11

to the non-computational state |20

and P_(l)=P₀₁(t_(p)) represents the leakage error.

FIG. 6 is a flow diagram of an example process 600 for generating acascade schedule used to implement an iSWAP quantum logic gate. Forexample, the process 600 may be used to generate the cascade schedulesfor implementing an iSWAP quantum logic gate between a first and secondqubit shown in plot 750 of FIG. 7. In the following, determining thecascade schedule will be described using “laboratory time”, t, though itmay alternatively be performed in “natural time”, τ.

The cascade schedule is generated based on a trapezoidal waveform for acontrol angle defining a trajectory of a frequency of the first qubitduring the gate execution. Generating the cascade schedule may firstinclude defining a trapezoidal ramp function (step 602)

${f_{r}\left( {t,s} \right)} = \left\{ \begin{matrix}\frac{t}{s \cdot t_{p}} & {0 \leq t \leq {s \cdot t_{p}}} \\1 & {{s \cdot t_{p}} < t < {\left( {1 - s} \right)t_{p}}} \\{1 - \frac{t - {\left( {1 - s} \right)t_{p}}}{s \cdot t_{p}}} & {{\left( {1 - s} \right)t_{p}} \leq t \leq t_{p}}\end{matrix} \right.$

where 0≤ƒ_(r)(t,s)≤1 and the dimensionless parameter s (0≤s≤½) controlsduration of the ramp-up (or ramp-down) intervals t_(r), such thatt_(r)=s·t_(p) and the plateau (or “wait”) time t_(W)=(1−2s)·t_(p). Thena trapezoidal waveform for the control angle may be generated using theramp function ƒ_(r)(t,s):

ϑ(t,s,θ _(max))=θ_(in)+(θ_(max)−θ_(in))ƒ_(r)(t,s).

Here θ_(in)=arccot[(ϵ(0)−η₁)/2√{square root over (2)}g], ϵ(0)≅1 GHzrepresents the initial value of the detuning, and θ_(max) represents themaximum angular distance travelled by the control vector. Therefore asimple trapezoidal waveform of the control angle is defined by the twocontrol parameters s and θ_(max). Determining specific values of theseparameters is described below with reference to step 608.

The number of the control parameters may be increased using polynomialexpansion of the waveform for the control angle in terms of thegenerated trapezoidal ramp function (step 604):

ϑ(t)=θ_(in)+(θ_(max)−θ_(in))[c ₁ƒ_(r)(t)+c ₂ƒ_(r) ²(t)+ . . . ]

where the coefficients of the polynomial expansion c₁, c₂, . . . sum toone. These coefficients along with s and θ_(max) can be obtained using aglobal optimization process using the fidelity as a fitness function (orgate error as a cost function.) In some implementations, one or moreconstraints may be applied to the coefficients {c}, such as a constraintthat the ramp function be smooth (i.e. the gradients match at theboundaries of the ramp).

An initial schedule for driving the detuning between energy levels ofthe first qubit and second qubit using the generated polynomialexpansion of the waveform for the control angle is determined (step 606)via

ε(t)=μ+2g·λ cot(ϑ,{c})]

with μ≅η₁ representing shift, λ≅√{square root over (2)} representingscaling, g representing inter-qubit interaction strength, ϑ representingthe control angle and {c} representing a set of two or more controlparameters, which comprise the relative ramp-up time s=t_(r)/t_(p) andthe height of the trapezoid θ_(max). as the parameters {c} may furthercomprise the set of the polynomial coefficients c₁, c₂, . . . . Thecomplete SWAP operation is possible if ϵ(t) crosses zero at the end ofthe second stage of the ramp-up (or ramp-down) region. This implies thatθ_(max) exceeds θ_(c)=π−arccot[η/2√{square root over (2)}g] with examplevalues of η≅200 MHz and g≅20 MHz, θ_(c)=0.91π. These values of θ_(max)are responsible for a characteristic shape of the cascade protocol.

One or more solutions in the swap channel are determined (step 608).This may include first determining a seed schedule with the minimal setof parameters s and θ_(max), which produces a complete SWAP operation ina targeted region of the gate times. Determining the seed schedule mayinclude calculating the unitary evolution matrix for the SWAP channel.Since the cascade schedule is based on the piecewise trapezoidalwaveform, the evolution matrix may be calculated by solving theSchrödinger equation within each time interval and matching thesolutions at the boundaries. By eliminating the state vectorscorresponding to the intermediate times, the solution is propagated fromthe beginning to the end of the execution. The unitary evolution matrixU(s,θ_(max)) describing SWAP channel is obtained as a product oftransfer matrices corresponding to each time interval. The transitionprobability P₀₁(s,θ_(max)) can be derived from U(s,θ_(max)) assumingthat the system is in the state |10

at t=0. The condition P₀₁(s,θ_(max))=1 ensures a complete populationswap between the qubits. Therefore, solving equation P₀₁(s,θ_(max))=1for one of the variables, say s, gives a “perfect swap trajectory”,s(θ_(max)), i.e. a path in a two-dimensional parameter space (s,θ_(max))for which P₀₁=1. This trajectory defines the seed schedule.

In some implementations a rounding Gaussian filter may be applied toeliminate high frequencies in the pulse Fourier spectrum and avoiddistortion of the pulse due to cutoff frequency in the electronic boards(DACers). In addition, the set of control parameters may be expanded byadding one or two higher order terms in the polynomial expansion of thecontrol angle to ensure greater variational freedom during theoptimization of the protocol.

The probability of an error in the leakage channel is determined (step610). Once the seed schedule is known the Schrödinger equation can besolved for the leakage channel, e.g., in a two-level approximation bykeeping only strongly interacting 11 and 20 states in the Hamiltonian ofequation (3). To solve the Schrödinger equation the followingrepresentation of the state vector in the leakage manifold can be used

|ψ_(l)(t)

=e ^(−iϕ) ^(l) ^((t))ϕ_(o)(t)|11

+e ^(iϕ) ^(l) ^((t))ϕ₁(t)|20

where ϕ_(l)(t)=(γ/2)∫₀ ^(2t/t) ^(p) dx({tilde over (ϵ)}(x)−{tilde over(η)}), {tilde over (ϵ)}(x)=ϵ(x)/2g and {tilde over (η)}=η/2g. Thefunctions ϕ_(o,1) can be found from the differential equations

4ϕ″₀(x)−4iγ({tilde over (ϵ)}(x)−{tilde over (η)})ϕ′₀(x)+2γ²ϕ₀(x)=0

4ϕ″₁(x)−4iγ({tilde over (ϵ)}(x)−{tilde over (η)})ϕ′₁(x)+2γ²ϕ₁(x)=0

and the initial conditions

${{\phi_{0}(0)} = {\cos\left( \frac{\theta_{l}}{2} \right)}},{\phi_{0}^{\prime{(0)}} = {{- \left( \frac{i\;\gamma}{\sqrt{2}} \right)}\mspace{14mu}\sin\mspace{14mu}\left( \frac{\theta_{l}}{2} \right)}},{{\phi_{1}(0)} = {\sin\left( \frac{\theta_{l}}{2} \right)}},{\phi_{1}^{\prime{(0)}} = {{- \left( \frac{i\;\gamma}{\sqrt{2}} \right)}{\cos\left( \frac{\theta_{l}}{2} \right)}}}$

where θ_(l)=arccot[({tilde over (ϵ)}(0)−{tilde over (η)})/√2]. Theleakage error P_(l) can then be calculated as

$\begin{matrix}{P_{l} = {{P_{11,20}\left( t_{p} \right)} = {{{{- e^{{- i}\;{\phi_{l}{(t_{p})}}}}\mspace{14mu}{\sin\left( \frac{\theta_{s}}{2} \right)}{\phi_{0}(2)}} + {e^{i\;{\phi_{l}{(t_{p})}}}{\cos\left( \frac{\theta_{s}}{2} \right)}{\phi_{1}(2)}}}}^{2}}} & (14)\end{matrix}$

The SWAP channel can be treated similarly using the representation|Ψ_(s)(t)

=e^(−iϕ) ^(s) ^((t))χ_(o)(t)|10

+e^(iϕ) ^(s) ^((t))χ₁(t)|01

in the swap manifold to obtain a SWAP error P_(s) given by

$\begin{matrix}{P_{s} = {{1 - {P_{10,01}\left( t_{p} \right)}} = {1 - {{{{- e^{{- i}\;{\phi_{s}{(t_{p})}}}}\mspace{14mu}{\sin\left( \frac{\theta_{s}}{2} \right)}{\chi_{0}(2)}} + {e^{i\;{\phi_{s}{(t_{p})}}}{\cos\left( \frac{\theta_{s}}{2} \right)}{\chi_{1}(2)}}}}^{2}}}} & (15)\end{matrix}$

The initial schedule for driving the detuning is adjusted bysynchronizing the one or more solutions in the swap channel and theprobability of an error in the leakage channel (step 662). Synchronizingthe solutions in the SWAP channel and the probability of an error in theleakage channel may include selecting a solution in the SWAP channelthat, when combined with the probability of an error in the leakagechannel, achieves acceptable swap gate fidelity. Values of parameters inthe selected solution may then be applied to the initial schedule fordriving the detuning to generate the cascade schedule. Examplesynchronizations of leakage and swap errors are illustrated in FIGS.9-11.

In some cases parameter values of the cascade schedule may be furtheradjusted using gate fidelity as a fitness function (or gate error as acost function.) To determine the fidelity of an iSWAP gate implementedusing the proposed Rosen-Zener cascade schedule, defined by the unitaryoperator U≡U(t_(p)) an ideal (or target) iSWAP gate U_(t) operating inthe entire two-qubit Hilbert space. The average fidelity F(U; U_(t)) canthen be calculated as

$\begin{matrix}{{F\left( {U;U_{t}} \right)} = {\frac{1}{d + 1}\left( {{\frac{1}{d^{2}}{\sum\limits_{j}{{Tr}\left( {U_{t}U_{j}^{\dagger}U_{t}^{\dagger}{UU}_{j}U^{\dagger}} \right)}}} + 1} \right)}} & (16)\end{matrix}$

where d represents the dimension of the Hilbert space, andU_(j)/√{square root over (d)} represents the orthonormal set of unitaryoperators, which forms a basis in the d²-dimensional space of d×dunitary matrices. The orthonormality condition is defined as TrU_(i)^(†)U_(j)=dδ_(ij). In this case, due to the conservation of the totalnumber of excitations, the state vector is confined within a6-dimensional space spanned by four computational states 00, 10, 01, 11and two non-computational states 20, 02. As such, the target iSWAP gatecan be defined as

$\begin{matrix}{U_{t} = \begin{pmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & {- i} & 0 & 0 & 0 \\0 & {- i} & 0 & 0 & 0 & 0 \\0 & 0 & 0 & e^{i\;\phi} & 0 & 0 \\0 & 0 & 0 & 0 & e^{{- i}\;\phi} & 0 \\0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}} & (17)\end{matrix}$

The basis {U_(j)} can be chosen. Usually it is chosen to be consistentwith the nature of actual physical systems that perform computations.Since physical systems performing computations are typically qutrits(three-level systems) rather than qubits in state of the art quantumcomputing architecture, a natural choice is to use direct products ofGell-Mann matrices or, since the Hilbert space is 6-dimensional, toutilize (properly normalized) generalized Gell-Mann matrices λ_(j) thatspan the Lie algebra of the SU(6) group. As an example, using the latteroption generates a 36-element set comprised of 35 matrices

$U_{j} = {\left( \frac{d}{2} \right)\lambda_{j}}$

plus the 6×6 unit matrix. Using this procedure and equations (16) and(17) obtains the following expression for the average fidelity:

$F = {\frac{1}{21}\left\lbrack {9 - {2\left( {P_{s} + P_{l}} \right)} + {4\left( {\sqrt{1 - P_{l}} + \sqrt{1 - P_{s}} + \sqrt{\left( {1 - P_{l}} \right)\left( {1 - P_{s}} \right)}} \right\rbrack}} \right.}$

where P_(s) and P_(l) are the SWAP and leakage errors defined above inequations (14) and (15). This expression for the fidelity can be used asa fitness function to adjust values of the parameters of the cascadeschedule.

Alternatively or in addition hardware testing and randomizedbenchmarking techniques may be performed to determine furtheradjustments to the generated cascade schedule, e.g., adjustments thatfurther increase the iSWAP gate fidelity.

The cascade schedule is then implemented using the generated schedule.Implementing the cascade schedule includes, during a first stage,adiabatically driving detuning between the frequency of the first qubitand the frequency of the second qubit through a first avoided crossingin a leakage channel; during a second stage, driving detuning betweenthe frequency of the first qubit and the frequency of the second qubitthrough a second avoided crossing in a swap channel; during a thirdstage, evolving the first qubit and second qubit; during a fourth stage,implementing the second stage in reverse order; and during a fifthstage, implementing the first stage in reverse order.

FIG. 7 shows two example cascade schedules generated using the process600 for implementing an iSWAP quantum logic gate between a first qubitand a second qubit described with reference to FIG. 6. The first plot700 in FIG. 7 shows two trapezoidal waveforms ϑ(t) In the firsttrapezoidal waveform 702 the parameters s and θ_(max) take values s=0.3and θ_(max)=0.917π. In the second trapezoidal waveform 704 theparameters s and θ_(max) take values s=0.4 and θ_(max)=0.91π. The secondplot 750 in FIG. 7 shows two cascade schedules ϵ(t) corresponding to thetwo trapezoidal ramp functions. The first cascade schedule 752corresponds to the first trapezoidal ramp function 702. The secondcascade schedule 754 corresponds to the second trapezoidal ramp function704. The values of μ and λ in both cascade schedules are set to μ=η,λ=√2. Plot 750 also shows avoided crossing in the SWAP channel (ϵ=0) 758and avoided crossing in the leakage channel (ϵ=η) 756. Representationsof the five stages described with reference to FIG. 6 are shown in FIG.7. However, these representations are example representations and forillustrative purposes only.

FIG. 8 shows a plot 800 of the probabilities of interqubit populationswap for the two example cascade schedules illustrated in FIG. 7. Line802 corresponds to probabilities of interqubit population swap for thecascade schedule with s=0.3 and θ_(max)=0.917π. Line 804 corresponds toprobabilities of interqubit population swap for the cascade schedulewith s=0.4 and θ_(max)=0.91π.

FIG. 8 shows how the behavior of Poi for both schedules drasticallydiffers in terms of the positions of the maxima, their width and generalshape. However, in both cases a complete SWAP operation is attainable.The general trend is that shrinking the trapezoidal base produces longergates, as expected. On the other hand, the longer gates are moreadiabatic and, in turn, more stable.

FIG. 9 shows an example synchronization of leakage and SWAP errors. Thedotted lines 902 represent gate times corresponding to zero leakage. Thehorizontal line 904 represents the gate time t_(p) such that P_(s)(t_(p))=0. As shown in FIG. 9, the gate time can be chosen to satisfyP_(s)(t_(p))≅P_(l)(t_(p))≅0 at the third zero of leakage error 906.

FIG. 10 shows an example synchronization of leakage and SWAP errors fora cascade schedule generated using a Rosen-Zener protocol with lowsensitivity to gate time error. The first plot 1000 shows exponentialdecay of the leakage error 1002. The second plot 1050 shows thecalculated fidelity of the gate enabled by the protocol. The calculatedfidelity shows a very low loss of fidelity. The seeming fidelity loss ispredominantly due to (also very small) deviation of the SWAP angle fromπ/2. Since the latter is not very important for the quantum supremacyexperiments this protocol could be a viable candidate for a fast anefficient hardware implementation.

Implementations of the digital and/or quantum subject matter and thedigital functional operations and quantum operations described in thisspecification can be implemented in digital electronic circuitry,suitable quantum circuitry or, more generally, quantum computationalsystems, in tangibly-embodied digital and/or quantum computer softwareor firmware, in digital and/or quantum computer hardware, including thestructures disclosed in this specification and their structuralequivalents, or in combinations of one or more of them. The term“quantum computational systems” may include, but is not limited to,quantum computers, quantum information processing systems, quantumcryptography systems, or quantum simulators.

Implementations of the digital and/or quantum subject matter describedin this specification can be implemented as one or more digital and/orquantum computer programs, i.e., one or more modules of digital and/orquantum computer program instructions encoded on a tangiblenon-transitory storage medium for execution by, or to control theoperation of, data processing apparatus. The digital and/or quantumcomputer storage medium can be a machine-readable storage device, amachine-readable storage substrate, a random or serial access memorydevice, one or more qubits, or a combination of one or more of them.Alternatively or in addition, the program instructions can be encoded onan artificially-generated propagated signal that is capable of encodingdigital and/or quantum information, e.g., a machine-generatedelectrical, optical, or electromagnetic signal, that is generated toencode digital and/or quantum information for transmission to suitablereceiver apparatus for execution by a data processing apparatus.

The terms quantum information and quantum data refer to information ordata that is carried by, held or stored in quantum systems, where thesmallest non-trivial system is a qubit, i.e., a system that defines theunit of quantum information. It is understood that the term “qubit”encompasses all quantum systems that may be suitably approximated as atwo-level system in the corresponding context. Such quantum systems mayinclude multi-level systems, e.g., with two or more levels. By way ofexample, such systems can include atoms, electrons, photons, ions orsuperconducting qubits. In many implementations the computational basisstates are identified with the ground and first excited states, howeverit is understood that other setups where the computational states areidentified with higher level excited states are possible.

The term “data processing apparatus” refers to digital and/or quantumdata processing hardware and encompasses all kinds of apparatus,devices, and machines for processing digital and/or quantum data,including by way of example a programmable digital processor, aprogrammable quantum processor, a digital computer, a quantum computer,multiple digital and quantum processors or computers, and combinationsthereof. The apparatus can also be, or further include, special purposelogic circuitry, e.g., an FPGA (field programmable gate array), an ASIC(application-specific integrated circuit), or a quantum simulator, i.e.,a quantum data processing apparatus that is designed to simulate orproduce information about a specific quantum system. In particular, aquantum simulator is a special purpose quantum computer that does nothave the capability to perform universal quantum computation. Theapparatus can optionally include, in addition to hardware, code thatcreates an execution environment for digital and/or quantum computerprograms, e.g., code that constitutes processor firmware, a protocolstack, a database management system, an operating system, or acombination of one or more of them.

A digital computer program, which may also be referred to or describedas a program, software, a software application, a module, a softwaremodule, a script, or code, can be written in any form of programminglanguage, including compiled or interpreted languages, or declarative orprocedural languages, and it can be deployed in any form, including as astand-alone program or as a module, component, subroutine, or other unitsuitable for use in a digital computing environment. A quantum computerprogram, which may also be referred to or described as a program,software, a software application, a module, a software module, a script,or code, can be written in any form of programming language, includingcompiled or interpreted languages, or declarative or procedurallanguages, and translated into a suitable quantum programming language,or can be written in a quantum programming language, e.g., QCL orQuipper.

A digital and/or quantum computer program may, but need not, correspondto a file in a file system. A program can be stored in a portion of afile that holds other programs or data, e.g., one or more scripts storedin a markup language document, in a single file dedicated to the programin question, or in multiple coordinated files, e.g., files that storeone or more modules, sub-programs, or portions of code. A digital and/orquantum computer program can be deployed to be executed on one digitalor one quantum computer or on multiple digital and/or quantum computersthat are located at one site or distributed across multiple sites andinterconnected by a digital and/or quantum data communication network. Aquantum data communication network is understood to be a network thatmay transmit quantum data using quantum systems, e.g. qubits. Generally,a digital data communication network cannot transmit quantum data,however a quantum data communication network may transmit both quantumdata and digital data.

The processes and logic flows described in this specification can beperformed by one or more programmable digital and/or quantum computers,operating with one or more digital and/or quantum processors, asappropriate, executing one or more digital and/or quantum computerprograms to perform functions by operating on input digital and quantumdata and generating output. The processes and logic flows can also beperformed by, and apparatus can also be implemented as, special purposelogic circuitry, e.g., an FPGA or an ASIC, or a quantum simulator, or bya combination of special purpose logic circuitry or quantum simulatorsand one or more programmed digital and/or quantum computers.

For a system of one or more digital and/or quantum computers to be“configured to” perform particular operations or actions means that thesystem has installed on it software, firmware, hardware, or acombination of them that in operation cause the system to perform theoperations or actions. For one or more digital and/or quantum computerprograms to be configured to perform particular operations or actionsmeans that the one or more programs include instructions that, whenexecuted by digital and/or quantum data processing apparatus, cause theapparatus to perform the operations or actions. A quantum computer mayreceive instructions from a digital computer that, when executed by thequantum computing apparatus, cause the apparatus to perform theoperations or actions.

Digital and/or quantum computers suitable for the execution of a digitaland/or quantum computer program can be based on general or specialpurpose digital and/or quantum processors or both, or any other kind ofcentral digital and/or quantum processing unit. Generally, a centraldigital and/or quantum processing unit will receive instructions anddigital and/or quantum data from a read-only memory, a random accessmemory, or quantum systems suitable for transmitting quantum data, e.g.photons, or combinations thereof.

The essential elements of a digital and/or quantum computer are acentral processing unit for performing or executing instructions and oneor more memory devices for storing instructions and digital and/orquantum data. The central processing unit and the memory can besupplemented by, or incorporated in, special purpose logic circuitry orquantum simulators. Generally, a digital and/or quantum computer willalso include, or be operatively coupled to receive digital and/orquantum data from or transfer digital and/or quantum data to, or both,one or more mass storage devices for storing digital and/or quantumdata, e.g., magnetic, magneto-optical disks, optical disks, or quantumsystems suitable for storing quantum information. However, a digitaland/or quantum computer need not have such devices.

Digital and/or quantum computer-readable media suitable for storingdigital and/or quantum computer program instructions and digital and/orquantum data include all forms of non-volatile digital and/or quantummemory, media and memory devices, including by way of examplesemiconductor memory devices, e.g., EPROM, EEPROM, and flash memorydevices; magnetic disks, e.g., internal hard disks or removable disks;magneto-optical disks; CD-ROM and DVD-ROM disks; and quantum systems,e.g., trapped atoms or electrons. It is understood that quantum memoriesare devices that can store quantum data for a long time with highfidelity and efficiency, e.g., light-matter interfaces where light isused for transmission and matter for storing and preserving the quantumfeatures of quantum data such as superposition or quantum coherence.

Control of the various systems described in this specification, orportions of them, can be implemented in a digital and/or quantumcomputer program product that includes instructions that are stored onone or more non-transitory machine-readable storage media, and that areexecutable on one or more digital and/or quantum processing devices. Thesystems described in this specification, or portions of them, can eachbe implemented as an apparatus, method, or system that may include oneor more digital and/or quantum processing devices and memory to storeexecutable instructions to perform the operations described in thisspecification.

While this specification contains many specific implementation details,these should not be construed as limitations on the scope of what may beclaimed, but rather as descriptions of features that may be specific toparticular implementations. Certain features that are described in thisspecification in the context of separate implementations can also beimplemented in combination in a single implementation. Conversely,various features that are described in the context of a singleimplementation can also be implemented in multiple implementationsseparately or in any suitable sub-combination. Moreover, althoughfeatures may be described above as acting in certain combinations andeven initially claimed as such, one or more features from a claimedcombination can in some cases be excised from the combination, and theclaimed combination may be directed to a sub-combination or variation ofa sub-combination.

Similarly, while operations are depicted in the drawings in a particularorder, this should not be understood as requiring that such operationsbe performed in the particular order shown or in sequential order, orthat all illustrated operations be performed, to achieve desirableresults. In certain circumstances, multitasking and parallel processingmay be advantageous. Moreover, the separation of various system modulesand components in the implementations described above should not beunderstood as requiring such separation in all implementations, and itshould be understood that the described program components and systemscan generally be integrated together in a single software product orpackaged into multiple software products.

Particular implementations of the subject matter have been described.Other implementations are within the scope of the following claims. Forexample, the actions recited in the claims can be performed in adifferent order and still achieve desirable results. As one example, theprocesses depicted in the accompanying figures do not necessarilyrequire the particular order shown, or sequential order, to achievedesirable results. In some cases, multitasking and parallel processingmay be advantageous.

1. A method for implementing an iSWAP quantum logic gate between a firstqubit and a second qubit, the method comprising: implementing a cascadeschedule that defines a trajectory of a detuning between a frequency ofthe first qubit and a frequency of the second qubit, comprising: duringa first stage, adiabatically driving detuning between the frequency ofthe first qubit and the frequency of the second qubit through a firstavoided crossing in a leakage channel; during a second stage, drivingdetuning between the frequency of the first qubit and the frequency ofthe second qubit through a second avoided crossing in a swap channel;during a third stage, evolving the first qubit and second qubit; duringa fourth stage, implementing the second stage in reverse order; andduring a fifth stage, implementing the first stage in reverse order. 2.The method of claim 1, wherein the cascade schedule satisfies a localadiabatic evolution condition.
 3. The method of claim 2, wherein thelocal adiabatic evolution condition is given by

Ψ_(g)(t)|∂_(t)Ψ_(e)(t)

=const·ω_(g)(t) or ∂_(t)θ(t)=const, where Ψ_(g)(t) and Ψ_(e)(t)represent instantaneous adiabatic eigenstates of an effectiveHamiltonian describing the leakage channel, θ(t) represents the controlangle, and ω_(g)(t)=√{square root over ((ϵ(t)−η₁)²+8g²)} represents atime dependent gap for the Hamiltonian describing the leakage channel,with ϵ(t) representing the detuning between the frequency of the firstqubit and the frequency of the second qubit, η₁ representing ananharmonicity parameter of the first qubit, and g representinginterqubit interaction strength.
 4. The method of claim 2, wherein theprobability that the cascade schedule incurs a leakage error isproportional to $e^{{- 2}\sqrt{2}{g \cdot t_{p}}}$ with g representingthe interqubit interaction strength and t_(p) representing a totalduration of the iSWAP gate.
 5. The method of claim 1, wherein thecascade schedule synchronizes minimal errors in the swap channel andleakage channel.
 6. The method of claim 1, wherein the first qubit andsecond qubit comprise capacitively coupled Xmon qubits.
 7. The method ofclaim 1, wherein the leakage channel comprises a manifold spanned by thecomputational state 11 and two non-computational states 02 and 20, andwherein driving detuning between the frequency of the first qubit andthe frequency of the second qubit through a first avoided crossing in aleakage channel comprises driving detuning between the frequency of thefirst qubit and the frequency of the second qubit through state 11-20resonance.
 8. The method of claim 1, wherein the swap channel comprisesa manifold spanned by the computational states 10 and 01, and whereindriving detuning between the frequency of the first qubit and thefrequency of the second qubit through a second avoided crossing in aswap channel comprises driving detuning between the frequency of thefirst qubit and the frequency of the second qubit through state 10-01resonance.
 9. The method of claim 1, wherein implementing the secondstage in reverse order comprises driving detuning between the frequencyof the first qubit and the frequency of the second qubit to achieve acomplete population swap between the qubit states 10 and
 01. 10. Themethod of claim 1, wherein the control angle comprises an angle betweenan effective magnetic field and the z-axis on a Bloch sphere of a systemcomprising the first qubit and the second qubit.
 11. The method of claim1, further comprising generating the cascade schedule, comprising:defining a trapezoidal ramp function in laboratory time; generating apolynomial expansion of the waveform for the control angle in terms ofthe generated trapezoidal ramp function; generating an initial schedulefor driving the detuning between the frequency of the first qubit andthe frequency of the second qubit using the generated polynomialexpansion of the waveform for the control angle; obtaining one or moresolutions in the swap channel; determining the probability of an errorin the leakage channel; and adjusting the initial schedule for drivingthe detuning by synchronizing the one or more solutions in the swapchannel and the probability of an error in the leakage channel,comprising: selecting a solution in the swap channel that, when combinedwith the probability of an error in the leakage channel, achieves apredetermined swap gate fidelity; and applying values of parameters inthe selected solution to the initial schedule for driving the detuningto generate the cascade schedule.
 12. The method of claim 11, furthercomprising adjusting parameter values of the cascade schedule using gatefidelity as a fitness function.
 13. The method of claim 11, furthercomprising performing hardware testing and randomized benchmarkingtechniques to adjust the generated cascade schedule.
 14. The method ofclaim 11, wherein the trapezoidal ramp function is a function of maximumcontrol angle, pulse duration, length of upward ramp, and length ofdownward ramp.
 15. The method of claim 11, wherein the trapezoidal rampfunction takes values that are less than or equal to one.
 16. The methodof claim 11, wherein coefficients of the polynomial expansion sum toone.
 17. The method of claim 11, wherein generating the initial scheduleoptionally further comprises applying a Gaussian filter.
 18. The methodof claim 11, wherein generating the initial schedule comprisesparameterizing the detuning as ϵ(t)=μ+2g·λ cot[Θ(t, {c})] with μrepresenting shift, λ representing scaling, g representing interqubitinteraction strength, Θ representing the generated polynomial expansionand c representing a set of two or more variational parameters.
 19. Anapparatus comprising: a first qubit; a second qubit coupled to the firstqubit; control electronics comprising one or more control devices thattune the frequency of the first qubit and second qubit throughapplication of respective control signals, wherein the controlelectronics are configured to implement a cascade schedule that definesa trajectory of a detuning between a frequency of the first qubit and afrequency of the second qubit, comprising: during a first stage,adiabatically driving detuning between the frequency of the first qubitand the frequency of the second qubit through a first avoided crossingin a leakage channel; during a second stage, driving detuning betweenthe frequency of the first qubit and the frequency of the second qubitthrough a second avoided crossing in a swap channel; during a thirdstage, evolving the first qubit and second qubit; during a fourth stage,implementing the second stage in reverse order; and during a fifthstage, implementing the first stage in reverse order.